DEFINITION:-
An ordered field F has the archimedean property if, given any positive x and y in F there is an integer n>0 so that nx>y.
Simply:-
If x,y €R and x,y>0 ,( y can be <0 ,) there is an integer n>0 so that nx>y.
PROOF:-
(by contradiction)
Let us assume that , nx<y ....................(1)
and nx is general element of set F
i. e,
F= {nx : x €R and x>0 , here n€N }
Since nx<y ( from assuming ...1)
(a) Thus y is an upper bound of set F.
(b) It shows set F is bounded above and a subset of R. Though completeness theorem we can say that there is also present a least upper bound ( or sup(F) ) ,, let we named it k....( say) .
Since;
x>0
Then, -x<0
By adding alpha both side,
-x+ alpha < alpha
alpha -x< alpha
Now if alpha is a supremum then obviously alpha -x is less than suprimum.
it also shows that alpha-x is not an upper bound of set F.
Let an other natural number m ,
For which,
alpha-x <mx
alpha<mx+x
alpha<x(m+1)
Since, m€ natural number then (m+1) is also a natural number .
Finally here,
alpha<x(m+1)
It shows that alpha is still small,
but we assumed that alpha is supremum of the set F.
THIS CONTRADICTION ARISES BECAUSE OF OUR WRONG ASSUMPTION 😑😒
THAT nx<y ,
Thus the correct property is given ,
In any positive x and y in F there is an integer n>0 so that nx>y.
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